Question

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 8 passengers per minute.

 a. Compute the probability of no arrivals in a one-minute period (to 6 decimals).

 b. Compute the probability that three or fewer passengers arrive in a one-minute period (to 4 decimals).

 c. Compute the probability of no arrivals in a 15-second period to 4 decimals).

 d. Compute the probability of at least one arrival in a 15-second period (to 4 decimals).

Answer 1

a)

Here, λ = 8 and x = 0
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X = 0)
P(X = 0) = 8^0 * e^-8/0!
P(X = 0) = 0.0003
Ans: 0.0003

b)
Here, λ = 8 and x = 3
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X <= 3).
P(X <= 3) = (8^0 * e^-8/0!) + (8^1 * e^-8/1!) + (8^2 * e^-8/2!) + (8^3 * e^-8/3!)
P(X <= 3) = 0.0003 + 0.0027 + 0.0107 + 0.0286
P(X <= 3) = 0.0423

c)

Here, λ = 2 and x = 0
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X = 0)
P(X = 0) = 2^0 * e^-2/0!
P(X = 0) = 0.1353
Ans: 0.1353

d)

Here, λ = 2 and x = 1
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X > = 1) = 1 - P(X <= 0).
P(X >= 1) = 1 - (2^0 * e^-2/0!)
P(X > = 1) = 1 - 0.1353
P(X > 1) = 0.8647